36 research outputs found
Kick stability in groups and dynamical systems
We consider a general construction of ``kicked systems''. Let G be a group of
measure preserving transformations of a probability space. Given its
one-parameter/cyclic subgroup (the flow), and any sequence of elements (the
kicks) we define the kicked dynamics on the space by alternately flowing with
given period, then applying a kick. Our main finding is the following stability
phenomenon: the kicked system often inherits recurrence properties of the
original flow. We present three main examples. 1) G is the torus. We show that
for generic linear flows, and any sequence of kicks, the trajectories of the
kicked system are uniformly distributed for almost all periods. 2) G is a
discrete subgroup of PSL(2,R) acting on the unit tangent bundle of a Riemann
surface. The flow is generated by a single element of G, and we take any
bounded sequence of elements of G as our kicks. We prove that the kicked system
is mixing for all sufficiently large periods if and only if the generator is of
infinite order and is not conjugate to its inverse in G. 3) G is the group of
Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the
flow is rapidly growing in the sense of Hofer's norm, and the kicks are
bounded. We prove that for a positive proportion of the periods the kicked
system inherits a kind of energy conservation law and is thus superrecurrent.
We use tools of geometric group theory and symplectic topology.Comment: Latex, 40 pages, revised versio
Shapes of polyhedra, mixed volumes and hyperbolic geometry
We generalize to higher dimensions the BavardâGhys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The AlexandrovâFenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to Ï/2
Biased belief updating and suboptimal choice in foraging decisions
Deciding which options to engage, and which to forego, requires developing accurate beliefs about the overall distribution of prospects. Here we adapt a classic prey selection task from foraging theory to examine how individuals keep track of an environmentâs reward rate and adjust choices in response to its fluctuations. Preference shifts were most pronounced when the environment improved compared to when it deteriorated. This is best explained by a trial-by-trial learning model in which participants estimate the reward rate with upward vs. downward changes controlled by separate learning rates. A failure to adjust expectations sufficiently when an environment becomes worse leads to suboptimal choices: options that are valuable given the environmental conditions are rejected in the false expectation that better options will materialize. These findings offer a previously unappreciated parallel in the serial choice setting of observations of asymmetric updating and resulting biased (often overoptimistic) estimates in other domains